# Chapter 3: Trigonometric Identities and Equations

**At Grade**Created by: CK-12

## Chapter Outline

- 3.1. Fundamental Identities
- 3.2. Proving Identities
- 3.3. Solving Trigonometric Equations
- 3.4. Sum and Difference Identities
- 3.5. Double Angle Identities
- 3.6. Half-Angle Identities
- 3.7. Products, Sums, Linear Combinations, and Applications

### Chapter Summary

## Chapter Summary

Here are the identities studied in this chapter:

Quotient & Reciprocal Identities

Pythagorean Identities

Even & Odd Identities

Co-Function Identities

Sum and Difference Identities

Double Angle Identities

Half Angle Identities

Product to Sum & Sum to Product Identities

Linear Combination Formula

, where and

## Review Questions

- Find the sine, cosine, and tangent of an angle with terminal side on .
- If and , find .
- Simplify: .
- Verify the identity:

For problems 5-8, find all the solutions in the interval .

- Solve the trigonometric equation over the interval .
- Solve the trigonometric equation over the interval .
- Solve the trigonometric equation for all real values of .

Find the exact value of:

- Write as a product:
- Simplify:
- Simplify:
- Derive a formula for .
- If you solve for , you would get . This new formula is used to reduce powers of cosine by substituting in the right part of the equation for . Try writing in terms of the first power of cosine.
- If you solve for , you would get . Similar to the new formula above, this one is used to reduce powers of sine. Try writing in terms of the first power of cosine.
- Rewrite in terms of the first power of cosine:

## Review Answers

- If the terminal side is on , then the hypotenuse of this triangle would be 17 (by the Pythagorean Theorem, ). Therefore, , and .
- If and , then is in Quadrant II. Therefore is negative. To find the third side, we need to do the Pythagorean Theorem. So .
- Factor top, cancel like terms, and use the Pythagorean Theorem Identity. Note that this simplification doesn't hold true for values of that are , where is a positive integer,, since the original expression is undefined for these values of .
- Change secant and cosecant into terms of sine and cosine, then find a common denominator.
- or radians and radians
- Because this is , you will need to divide by 3 at the very end to get the final answer. This is why we went beyond the limit of when finding .
- Rewrite the equation in terms of tan by using the Pythagorean identity, . Because these factors are the same, we only need to solve one for . Where is any integer.
- Use the half angle formula with .
- Use the sine sum formula.
- Use the sine and cosine sum formulas.
- Use the sine sum formula as well as the double angle formula.
- Using our new formula, Now, our final answer needs to be in the first power of cosine, so we need to find a formula for . For this, we substitute everywhere there is an and the formula translates to . Now we can write in terms of the first power of cosine as follows.
- Using our new formula, Now, our final answer needs to be in the first power of cosine, so we need to find a formula for . For this, we substitute everywhere there is an and the formula translates to . Now we can write in terms of the first power of cosine as follows.
- (a) First, we use both of our new formulas, then simplify: (b) For tangent, we use the identity and then substitute in our new formulas. Now, use the formulas we derived in #18 and #19.

## Texas Instruments Resources

*In the CK-12 Texas Instruments Trigonometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9701.*